Essentially, all a delegate is, is an object that accepts feedback from another object. Put simply, when stuff happens to an object, it tells its delegate (assuming it has one).
For instance, lets say I have a UIViewController with a UITextView placed in the middle of the view. I set up my UIViewController to be the delegate of the UITextView. Then, when certain actions are performed on the text view (begin editing, text changes, end editing, etc), it tells it's delegate so it can do whatever logic it needs to do, like spell checking every time characters change, or dismissing the keyboard when it receives a return key press.
Delegate methods perform a similar function to callback functions in C.
Hope that makes sense :)